Analytic construction of dimension-dependent steering witnesses from generalized equiangular measurements

Abstract

We construct a dimension-dependent family of generalized equiangular measurements that gives explicit steering witnesses in every prime dimension $d\geq5$. The construction consists of $d-1$ mutually unbiased measurements and an additional normalized family of $2d-1$ Hermitian operators depending on a parameter $τ$. Positivity of the latter family is the main technical problem. We show that the noncentral operators reduce to three-dimensional invariant blocks, while the remaining central operator is exactly solvable in a two-dimensional subspace. Two unitarily equivalent $3\times3$ matrix representations describe the same active spectrum, but yield different Gershgorin enclosures because Gershgorin bounds are basis dependent. Combining the stronger endpoint from each representation gives a fully analytic asymmetric interval for $τ$ that guarantees a valid POVM. Combining this POVM with the $d-1$ MUMs yields an explicit GEAM. Finally, using a maximally entangled realization and the local-hidden-state bound previously derived for generalized equiangular measurements, we obtain an explicit steering inequality and a sufficient criterion guaranteeing quantum violation in every prime dimension $d\geq5$.